## Work Rate Problems with Solutions

A set of problems related to work and rate of work is presented with detailed solutions.

## 9.10 Rate Word Problems: Work and Time

If it takes Felicia 4 hours to paint a room and her daughter Katy 12 hours to paint the same room, then working together, they could paint the room in 3 hours. The equation used to solve problems of this type is one of reciprocals. It is derived as follows:

$\text{rate}\times \text{time}=\text{work done}$

For this problem:

$\begin{array}{rrrl} \text{Felicia's rate: }&F_{\text{rate}}\times 4 \text{ h}&=&1\text{ room} \\ \\ \text{Katy's rate: }&K_{\text{rate}}\times 12 \text{ h}&=&1\text{ room} \\ \\ \text{Isolating for their rates: }&F&=&\dfrac{1}{4}\text{ h and }K = \dfrac{1}{12}\text{ h} \end{array}$

To make this into a solvable equation, find the total time $(T)$ needed for Felicia and Katy to paint the room. This time is the sum of the rates of Felicia and Katy, or:

$\begin{array}{rcrl} \text{Total time: } &T \left(\dfrac{1}{4}\text{ h}+\dfrac{1}{12}\text{ h}\right)&=&1\text{ room} \\ \\ \text{This can also be written as: }&\dfrac{1}{4}\text{ h}+\dfrac{1}{12}\text{ h}&=&\dfrac{1 \text{ room}}{T} \\ \\ \text{Solving this yields:}&0.25+0.083&=&\dfrac{1 \text{ room}}{T} \\ \\ &0.333&=&\dfrac{1 \text{ room}}{T} \\ \\ &t&=&\dfrac{1}{0.333}\text{ or }\dfrac{3\text{ h}}{\text{room}} \end{array}$

Example 9.10.1

Karl can clean a room in 3 hours. If his little sister Kyra helps, they can clean it in 2.4 hours. How long would it take Kyra to do the job alone?

The equation to solve is:

$\begin{array}{rrrrl} \dfrac{1}{3}\text{ h}&+&\dfrac{1}{K}&=&\dfrac{1}{2.4}\text{ h} \\ \\ &&\dfrac{1}{K}&=&\dfrac{1}{2.4}\text{ h}-\dfrac{1}{3}\text{ h}\\ \\ &&\dfrac{1}{K}&=&0.0833\text{ or }K=12\text{ h} \end{array}$

Example 9.10.2

Doug takes twice as long as Becky to complete a project. Together they can complete the project in 10 hours. How long will it take each of them to complete the project alone?

$\begin{array}{rrl} \dfrac{1}{R}+\dfrac{1}{2R}&=&\dfrac{1}{10}\text{ h,} \\ \text{where Doug's rate (} \dfrac{1}{D}\text{)}& =& \dfrac{1}{2}\times \text{ Becky's (}\dfrac{1}{R}\text{) rate.} \\ \\ \text{Sum the rates: }\dfrac{1}{R}+\dfrac{1}{2R}&=&\dfrac{2}{2R} + \dfrac{1}{2R} = \dfrac{3}{2R} \\ \\ \text{Solve for R: }\dfrac{3}{2R}&=&\dfrac{1}{10}\text{ h} \\ \text{which means }\dfrac{1}{R}&=&\dfrac{1}{10}\times\dfrac{2}{3}\text{ h} \\ \text{so }\dfrac{1}{R}& =& \dfrac{2}{30} \\ \text{ or }R &= &\dfrac{30}{2} \end{array}$

This means that the time it takes Becky to complete the project alone is $15\text{ h}$.

Since it takes Doug twice as long as Becky, the time for Doug is $30\text{ h}$.

Example 9.10.3

Joey can build a large shed in 10 days less than Cosmo can. If they built it together, it would take them 12 days. How long would it take each of them working alone?

$\begin{array}{rl} \text{The equation to solve:}& \dfrac{1}{(C-10)}+\dfrac{1}{C}=\dfrac{1}{12}, \text{ where }J=C-10 \\ \\ \text{Multiply each term by the LCD:}&(C-10)(C)(12) \\ \\ \text{This leaves}&12C+12(C-10)=C(C-10) \\ \\ \text{Multiplying this out:}&12C+12C-120=C^2-10C \\ \\ \text{Which simplifies to}&C^2-34C+120=0 \\ \\ \text{Which will factor to}& (C-30)(C-4) = 0 \end{array}$

Cosmo can build the large shed in either 30 days or 4 days. Joey, therefore, can build the shed in 20 days or −6 days (rejected).

The solution is Cosmo takes 30 days to build and Joey takes 20 days.

Example 9.10.4

Clark can complete a job in one hour less than his apprentice. Together, they do the job in 1 hour and 12 minutes. How long would it take each of them working alone?

$\begin{array}{rl} \text{Convert everything to hours:} & 1\text{ h }12\text{ min}=\dfrac{72}{60} \text{ h}=\dfrac{6}{5}\text{ h}\\ \\ \text{The equation to solve is} & \dfrac{1}{A}+\dfrac{1}{A-1}=\dfrac{1}{\dfrac{6}{5}}=\dfrac{5}{6}\\ \\ \text{Therefore the equation is} & \dfrac{1}{A}+\dfrac{1}{A-1}=\dfrac{5}{6} \\ \\ \begin{array}{r} \text{To remove the fractions, } \\ \text{multiply each term by the LCD} \end{array} & (A)(A-1)(6)\\ \\ \text{This leaves} & 6(A)+6(A-1)=5(A)(A-1) \\ \\ \text{Multiplying this out gives} & 6A-6+6A=5A^2-5A \\ \\ \text{Which simplifies to} & 5A^2-17A +6=0 \\ \\ \text{This will factor to} & (5A-2)(A-3)=0 \end{array}$

The apprentice can do the job in either $\dfrac{2}{5}$ h (reject) or 3 h. Clark takes 2 h.

Example 9.10.5

A sink can be filled by a pipe in 5 minutes, but it takes 7 minutes to drain a full sink. If both the pipe and the drain are open, how long will it take to fill the sink?

The 7 minutes to drain will be subtracted.

$\begin{array}{rl} \text{The equation to solve is} & \dfrac{1}{5}-\dfrac{1}{7}=\dfrac{1}{X} \\ \\ \begin{array}{r} \text{To remove the fractions,} \\ \text{multiply each term by the LCD}\end{array} & (5)(7)(X)\\ \\ \text{This leaves } & (7)(X)-(5)(X)=(5)(7)\\ \\ \text{Multiplying this out gives} & 7X-5X=35\\ \\ \text{Which simplifies to} & 2X=35\text{ or }X=\dfrac{35}{2}\text{ or }17.5 \end{array}$

17.5 min or 17 min 30 sec is the solution

For Questions 1 to 8, write the formula defining the relation. Do Not Solve!!

• Bill’s father can paint a room in 2 hours less than it would take Bill to paint it. Working together, they can complete the job in 2 hours and 24 minutes. How much time would each require working alone?
• Of two inlet pipes, the smaller pipe takes four hours longer than the larger pipe to fill a pool. When both pipes are open, the pool is filled in three hours and forty-five minutes. If only the larger pipe is open, how many hours are required to fill the pool?
• Jack can wash and wax the family car in one hour less than it would take Bob. The two working together can complete the job in 1.2 hours. How much time would each require if they worked alone?
• If Yousef can do a piece of work alone in 6 days, and Bridgit can do it alone in 4 days, how long will it take the two to complete the job working together?
• Working alone, it takes John 8 hours longer than Carlos to do a job. Working together, they can do the job in 3 hours. How long would it take each to do the job working alone?
• Working alone, Maryam can do a piece of work in 3 days that Noor can do in 4 days and Elana can do in 5 days. How long will it take them to do it working together?
• Raj can do a piece of work in 4 days and Rubi can do it in half the time. How long would it take them to do the work together?
• A cistern can be filled by one pipe in 20 minutes and by another in 30 minutes. How long would it take both pipes together to fill the tank?

For Questions 9 to 20, find and solve the equation describing the relationship.

• If an apprentice can do a piece of work in 24 days, and apprentice and instructor together can do it in 6 days, how long would it take the instructor to do the work alone?
• A carpenter and his assistant can do a piece of work in 3.75 days. If the carpenter himself could do the work alone in 5 days, how long would the assistant take to do the work alone?
• If Sam can do a certain job in 3 days, while it would take Fred 6 days to do the same job, how long would it take them, working together, to complete the job?
• Tim can finish a certain job in 10 hours. It takes his wife JoAnn only 8 hours to do the same job. If they work together, how long will it take them to complete the job?
• Two people working together can complete a job in 6 hours. If one of them works twice as fast as the other, how long would it take the slower person, working alone, to do the job?
• If two people working together can do a job in 3 hours, how long would it take the faster person to do the same job if one of them is 3 times as fast as the other?
• A water tank can be filled by an inlet pipe in 8 hours. It takes twice that long for the outlet pipe to empty the tank. How long would it take to fill the tank if both pipes were open?
• A sink can be filled from the faucet in 5 minutes. It takes only 3 minutes to empty the sink when the drain is open. If the sink is full and both the faucet and the drain are open, how long will it take to empty the sink?
• It takes 10 hours to fill a pool with the inlet pipe. It can be emptied in 15 hours with the outlet pipe. If the pool is half full to begin with, how long will it take to fill it from there if both pipes are open?
• A sink is ¼ full when both the faucet and the drain are opened. The faucet alone can fill the sink in 6 minutes, while it takes 8 minutes to empty it with the drain. How long will it take to fill the remaining ¾ of the sink?
• A sink has two faucets: one for hot water and one for cold water. The sink can be filled by a cold-water faucet in 3.5 minutes. If both faucets are open, the sink is filled in 2.1 minutes. How long does it take to fill the sink with just the hot-water faucet open?
• A water tank is being filled by two inlet pipes. Pipe A can fill the tank in 4.5 hours, while both pipes together can fill the tank in 2 hours. How long does it take to fill the tank using only pipe B?

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## How to Solve Combined Labor Problems

Last Updated: September 19, 2023

This article was co-authored by wikiHow Staff . Our trained team of editors and researchers validate articles for accuracy and comprehensiveness. wikiHow's Content Management Team carefully monitors the work from our editorial staff to ensure that each article is backed by trusted research and meets our high quality standards. This article has been viewed 70,267 times. Learn more...

Combined labor problems, or work problems, are math problems involving rational equations. [1] X Research source These are equations that involve at least one fraction. The problems basically require finding unit rates, combining them, and setting them equal to an unknown rate. These problems require a lot of interpretive logic, but as long as you know how to work with fractions, solving them is fairly easy.

## Problems with Two People Working Together

• For example, the problem might ask, “If Tommy can paint a room in 3 hours, and Winnie can paint the same room in 4 hours, how long will it take them to paint the room together?

## Problems with Two People Working Against Each Other

• For example, the problem might ask, “If a hose can fill a pool 6 hours, and an open drain can empty it in 2 hours, how long will it take the open drain to empty the pool with the hose on?”

## Problems with Two People Working In Shifts

• For example, the problem might be: “Damarion can clean the cat shelter in 8 hours, and Cassandra can clean the shelter in 4 hours. They work together for 2 hours, but then Cassandra leaves to take some cats to the vet. How long will it take for Damarion to finish cleaning the shelter on his own?”

## Video . By using this service, some information may be shared with YouTube.

• Pay close attention to units. These methods will work for any unit of time, such as minutes or days. Some problems might state the rates in different units, and you will need to convert. Thanks Helpful 0 Not Helpful 0
• If the problem involves more than two workers, simply add their individual work rates, then take the reciprocal of the sum to get the time taken working together. Thanks Helpful 0 Not Helpful 0

• A calculator

## You Might Also Like

• ↑ http://www.mathguide.com/lessons/Word-Work.html
• ↑ http://www.algebralab.org/Word/Word.aspx?file=Algebra_WorkingTogether.xml
• ↑ https://www.mtsac.edu/marcs/worksheet/math51/course/10application_problems_rational_expressions.pdf
• ↑ http://purplemath.com/modules/workprob2.htm

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## GMAT Work Rate Problems

GMAT rate problems might seem intimidating, but they’re not so bad once you get the fundamental concepts down. We’ll show you how to master this common type of GMAT word problem in this post, then give you some practice problems with answers and explanations!

## GMAT Rate and Work Rate Problems: Main Concepts

Big idea #1: the “art” equation.

You may be familiar with the distance equation, D = RT (“distance equals rate times time”), sometimes remembered as the “dirt” equation. It turns out, that equation is just a specific instance of a much more general equation. In that equation, R, the rate, is distance per time, but in non-distance problems, rate can be anything over time — wrenches produced per hour, houses painted per day, books written per decade, etc. In these cases, typical of work problems, we are no longer concerned with “distance” per time, but with the amount of something produced per time. We use A to represent this amount (the number of wrenches, the number of houses, etc.), and the equation becomes A = RT. Sometimes folks remember this as the “art” equation.

Here’s a simple mnemonic. When you travel, you are moving on the Earth, which is made of dirt, so for traveling & distance you use D = RT. Work problems involve machines, and machines make things —– making is creation, and creation is the essence of art, so use the A = RT equation. (I know, I know, what comes out of most machines is hardly worthy of aesthetic elevation, but it works for a mnemonic!)

## Big Idea #2: Rates are Ratios

The word “rate” and the word “ratio” have the same Latin root: in fact, they also share a Latin root with the “rationality” of our minds, but that’s a discussion that would bring up to our noses into Pythagorean and Platonic philosophies. The point is: a rate is a ratio, that is to say, a fraction. Technically, any fraction, any ratio, in which the numerator and the denominator have different units is a rate. Fuel efficiency (mpg) and price per unit and most baseball fractions (ERA, BA, OBP, SLG, etc.) are rates. Currency rates and exchanges rates are common financial market rates that, ironically, almost never appear on the GMAT —- go figure! Most GMAT rates have time in the denominator, and it’s a rate of how fast work is being done or how fast something is being produced or accomplished.

The fact that rates are ratios means: we can solve these problems by setting up proportions and using proportional thinking! As you will see in the solutions below, that’s an extremely powerful strategy for solution.

## Big Idea #3: Add Rates

The vast majority of work problems on the GMAT involve two people or two machines and comparisons of their individual production to their combined production. The questions will often give you information about times and about amounts, and what you need to know is: you can’t add or subtract times to complete a job and you can’t add or subtract amounts of work; instead, you add and subtract rates .

(rate of A alone) + (rate of B alone) = (combined rate of A & B)

Here A and B can be two people, two machines, etc. The extension of this idea is that if you have N identical machines, and each one works at a rate of R, then the combined rate is N*R.

## Big Idea #4: Understand Speed and Average Speed

Rate is another word for speed. One common source of errors with GMAT rate problems is that all three variables have to be in the same units. If you travel at 30 mph for 10 minutes, you do not go 30*10 = 300 miles!

Here’s an example:

Cross-multiply, and you get 10 = 2x. So, x must equal 5 miles.

Many trickier rate questions ask about “average speed” or “average velocity” (for GMAT purposes, those two are identical). The formula for average speed is:

$$\text{Average speed}=\frac {\text{Total distance}}{\text{Total time}}$$   For a single trip at one speed, there’s nothing particularly mysterious about this question. This concept becomes much trickier in two-leg trips, especially trips in which the car travels at one speed in one leg, and at another speed in another leg. You can never simply average the two velocities given, and that will always be a tempting incorrect choice on the GMAT. You always need to apply D = RT separately in each leg of the trip, and then you need to add results from the individual legs to find the total distance and the total time.

With just these four ideas, you can unlock any GMAT work rate problem. At this point, you may want to go back and give another attempt at those three practice questions. Follow carefully how they are applied in the solutions below.

## Practice GMAT Rate Problems

For practice, here are some GMAT rate problems for you! The last two are challenging.

A. 36 B. 40 C. 42 D. 45 E. 57

In order to figure out the average velocity, we need to know both the total distance and the total time. From the question, we know the total distance is 600 miles. We need to figure out the time of each leg separately. In the first leg, T = D/R = 300/30 = 10 hr. In the second leg, T = D/R = 300/60 = 5 hours. The total time is 10 + 5 = 15 hours. The average velocity, total distance divided by total time, is 600/15 = 40 mph. Answer = B. 2) A car drives for 3 hours at 40 mph and then drives 300 miles at 60 mph. What is the car’s average speed, in mph?

A. 45 B. 47.5 C. 50 D. 52.5 E. 55

In the first leg, we know time and rate, so find distance: D = RT = (3)*(40) = 120 miles. In the second leg, we know distance and rate, so find time: T = R/D = 300/60 = 5 hours. Total distance = 120 + 300 = 420 miles. Total time = 3 + 5 = 8. Average velocity = 420/8 = 210/4 = 105/2 = 52.5 mph. Answer = D . 3) For the first 150 miles of a trip, a car drives at v mph. For the next 200 miles, the car drives at (v + 25) mph. The average speed of the whole trip is 35 mph. Find the value of v.

A. 20 B. 25 C. 30 D. 35 E. 40

The distance of the first leg is 150 miles, and the rate is v, so the time of the first leg is:

$$\text{t}_1 = \frac{150}{\text{v}}$$   The distance for the second leg is 200, and the rate is v+25, so the time of the second leg is:

$$\text{t}_2 = \frac{200}{\text{v}+25}$$   The total distance was 350 miles, and the average speed was 35 mph, so the total time of the trip must have been T = D/R = 350/35 = 10 hours. At this point, the algebra becomes hairy, so I will just plug in numbers from the answer choices.

Choice A. If v = 20 mph, then v + 25 = 45 mph. The first leg takes 150/20 = 7.5 hours, and the last leg 200/45 takes way more than three hours, so this total time is well over 10 hours. This choice is not correct.

Choice B. If v = 25, then v + 25 = 50. The first leg takes 150/25 = 6 hours. The second leg takes 200/50 = 4 hours. The total is 10 hours, which is the correct value, so this is the correct answer choice. Answer = B . 4) A car travels at one speed for 4 hours, and then at twice that speed for 6 hours. The average velocity for the whole 10-hour trip is 40 mph. Find the initial speed in mph.

A. 25 B. 35 C. 40 D. 50 E. 60

If the average velocity for the 10 hour trip is 40 mph, that means the total distance is D = RT = (40)*(10) = 400 miles. The distance in the first leg is d 1 = RT = 4v. The distance in the second leg is d 2 = RT = (2v)*(6) = 12v. The total distance is the sum, 4v + 12v = 16 v. Set this equal to the numerical value of the total distance.

400 = 16v → 100 = 4v → 25 = v

So the initial speed is v = 25 mph. Answer = A . 5) Running at the same rate, 8 identical machines can produce 560 paperclips a minute. At this rate, how many paperclips could 20 machines produce in 6 minutes?

(A) 1344 (B) 3360 (C) 8400 (D) 50400 (E) 67200

“Running at the same rate, 8 identical machines can produce 560 paperclips a minute.” That 560 is a combined rate of 8 machines —- 560 = 8*R, so the rate of one machine is R = 560/8 = 70 paperclips per minute.

“At this rate, how many paperclips could 20 machines produce in 6 minutes?” Well, the combined rate of 20 machines would be Rtotal = 20*70 = 1400 pc/min. Now, plug that into the “art” equation: A = RT = (1400)*(6) = 8400 pc. Answer = C. 6) Jane can make a handcrafted drum in 4 weeks. Zane can make a similar handcrafted drum in 6 weeks. If they both work together, how many weeks will it take for them to produce 15 handcrafted drums?

(A) 30 (B) 36 (C) 70 (D) 80 (E) 150

Method I : the rates solution

“Jane can make a handcrafted drum in 4 weeks. Zane can make a similar handcrafted drum in 6 weeks.” Jane’s rate is (1 drum)/(4 weeks) = 1/4. Zane’s rate is (1 drum)/(6 weeks) = 1/6. The combined rate of Jane + Zane is

R = 1/4 + 1/6 = 3/12 + 2/12 = 5/12

That’s the combined rate. We need to make 15 drums — we have a rate and we have an amount, so use the “art” equation to solve for time:

T = A/R = 15/(5/12) = 15*(12/5) = (15/5)*12 = 3*12 = 36

BTW, notice in the penultimate step, the universal fraction strategy: cancel before you multiply (Tip #3: https://magoosh.com/gmat/2012/can-i-use-a-calculator-on-the-gmat/ . Jane and Zane need 36 weeks to make 15 drums. Answer = B .

Method II: the proportion solution

“Jane can make a handcrafted drum in 4 weeks. Zane can make a similar handcrafted drum in 6 weeks.” Let’s find the LCM of 4 and 6 — that’s 12 weeks. In a 12 week period, Jane, making a drum every 4 weeks, makes three drums. In a 12 week period, Zane, making a drum every 6 weeks, makes two drums. Therefore, in a 12 weeks period, they produce 5 drums between the two of them. If they make 5 drums in 12 weeks, they need triple that time, 36 weeks, to make 15 drums. Therefore, Jane and Zane need 36 weeks to make 15 drums. Answer = B . 7) Machines P and Q are two different machines that cover jars in a factory. When Machine P works alone, it covers 1500 jars in m hours. When Machines P and Q work simultaneously at their respective rates, they cover 1500 jars in n hours. In terms of m and n, how many hours does it take Machine Q, working alone at its constant rate, to cover 1500 jars?

(A) m/(m + n) (B) n/(m + n) (C) mn/(m + n) (D) mn/(m – n) (E) mn/(n – m)

This is a particularly challenging, one because we have variables in the answer choices. I will show an algebraic solution, although a numerical solution ( https://magoosh.com/gmat/math/word-problems/variables-in-gmat-answer-choices-algebraic-approach-vs-numerical-approach/ ) is always possible.

“Machines P and Q are two different machines that cover jars in a factory. When Machine P works alone, it covers 1500 jars in m hours. When Machines P and Q work simultaneously at their respective rates, they cover 1500 jars in n hours. In terms of m and n, how many hours does it take Machine Q, working alone at its constant rate, to cover 1500 jars? ”

Since the number “1500 jars” appears over and over, let’s arbitrarily say 1500 jars = 1 lot, and we’ll use units of lots per hour to simplify our calculations.

P’s individual rate is (1 lot)/(m hours) = 1/m. The combined rate of P & Q is (1 lot)/(n hours) = 1/n. We know

(P’s rate alone) + (Q’s rate alone) = (P and Q’s combined rate)

(Q’s rate alone) = (P and Q’s combined rate) – (P’s rate alone)

(Q’s rate alone) = 1/n – 1/m = m/ (nm) – n/ (nm) = (m – n)/(nm)

We now know Q’s rate, and we want the amount of 1 lot, so we use the “art” equation.

1 = [(m – n)/ (nm)]*T

T = (mn)/(m – n)

Answer = D 8) Working together, 7 identical pumps can empty a pool in 6 hours. How many hours will it take 4 pumps to empty the same pool?

(A) $$4 \frac{2}{3}$$ (B) $$9 \frac{1}{4}$$ (C) $$9 \frac{1}{3}$$ (D) $$9 \frac{3}{4}$$ (E) $$10 \frac{1}{2}$$

That’s all there is to GMAT work rate problems! Do you still have questions? Leave a comment below!

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Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as "member of the month" for over two years at GMAT Club . Mike holds an A.B. in Physics (graduating magna cum laude ) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's Youtube video explanations and resources like What is a Good GMAT Score? and the GMAT Diagnostic Test .

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## 12 Responses to GMAT Work Rate Problems

I’m getting lost at this line… can anyone help me understand how this logically makes sense:

(Q’s rate alone) = 1/n – 1/m = m/ (nm) – n/ (nm) = (m – n)/(nm)

So, let’s first briefly walk through what we know. We know that “(P’s rate alone) + (Q’s rate alone) = (P and Q’s combined rate).” We are also told that “(P’s rate alone) = (1/m)”, while “(P and Q’s combined rate) = (1/n).” Now, if we plug this information in and solve for “(Q’s rate alone)”, we get the following equation:

(Q’s rate alone) = (P and Q’s combined rate) – (P’s rate alone) (Q’s rate alone) = 1/n – 1/m

Next, if we want to add these two fractions together, both fractions require the same denominator. So, we’ll have to multiply “m” to numerator and denominator of “1/n”. We will also multiply “n” to the numerator and denominator of “1/m”.

(Q’s rate alone) = m/(nm) – n/(nm)

Remember, we aren’t changing these two individual fractions (i.e., (1/n) and (1/m)). We are just rephrasing them. For example, “m/(nm) = 1/n” if we cancel out the “m” in the numerator and denominator. Okay, now that the two fractions have the same denominator, we can subtract them together.

(Q’s rate alone) = (m-n)/(nm)

Hope this helps!

Your theories and concepts always help me alot , be it any topic!! Thanks Mike :’)

I had a quick question about that one video where you mention you CANNOT cross simplify with proportions. For a specific question on the video: “A machine can produce 36 staplers in 28 minutes, how many staplers can it produce in 1 hour and 45 minutes?”

Here is what I did, and I arrived at the correct answer. I set the equation up as you did initially, but instead of using a proportion, I cross multiplied between 9/7 and 105/1. Since we are making 9 staplers every 7 minutes, how many staplers would we make in 145 minutes. Thus I just multiplied using these two numbers and in fact I did simplify 7 and 105 to get 15. From there 15(9) = 135. The next thing you mentioned is not to cross multiply so I got concerned whether this approach was feasible or not. Did I just get lucky or is setting up a proportion totally different?

Are these enough for all WRT problems that appear on GMAT?

Dear Ravi, I’m glad you found this helpful. 🙂 I’m going to give you a piece of advice. Don’t ask the question “Are these enough?” or “Is this enough?” or “Can I consider myself done after this?” Those are the questions of mediocrity, and they consistently lead to limited performances. Consider the questions “What else can I understand about this?” and “How can I understand this topic more deeply?” and “What else can I do to improve myself?” Those are the questions of excellence. When you follow those questions and live by them, you are able to bring your best to any challenge. Does all this make sense? Mike 🙂

I know you cover frequency of GMAT topics on this website, but in terms of the OG 13 how many 700+ level questions does it have for the Quant and DS respectively? Also I see people posting Quant Scores of 51 how is this possible if there is only 37 Quant Questions?

Dear Sagnik, First of all, please understand that the entire idea of a “700+” question is very vague, not at all well defined. Roughly, it means questions that are among the hardest one might see on the GMAT. To know the exact level of any particular question, we would have to know the percentage of folks who get the question correct. If fewer than 10% got the question correct, then maybe it could be called a “700+” question. I don’t think many questions in the OG fit this description — maybe 10% or 15% of the hardest ones. Also, I think you need to understand a little more about how the GMAT is scored. See these two posts: https://magoosh.com/gmat/2012/what-is-the-gmat-cat-computer-adaptive-test/ https://magoosh.com/gmat/2013/gmat-score-percentiles/ The score of 51 on Quant is a scaled score that represents a percentile rank. What matters is not simply how many questions one gets right, but the difficulty level of each question. The computer does a very complicated calculation to get from one’s individual correct & incorrect questions to that scaled score. Does all this make sense? Mike 🙂

Thank you!! I´ve been struggling with this type of questions and with your explanation I could finally solve them! I do the Gmat in one week!

Maria: Thank you for your kind words. Best of luck to you. Mike 🙂

Thanks for the post. That was extremely helpful! Undoubtedly it will get me through these type of questions much faster now. Best,

Fernanda, I am very glad you found this helpful. Best of luck to you! Mike 🙂

Magoosh blog comment policy : To create the best experience for our readers, we will only approve comments that are relevant to the article, general enough to be helpful to other students, concise, and well-written! 😄 Due to the high volume of comments across all of our blogs, we cannot promise that all comments will receive responses from our instructors. We highly encourage students to help each other out and respond to other students' comments if you can! If you are a Premium Magoosh student and would like more personalized service from our instructors, you can use the Help tab on the Magoosh dashboard. Thanks!

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## "Work" Word Problems

Painting & Pipes Tubs & Man-Hours Unequal Times Etc.

"Work" problems usually involve situations such as two people working together to paint a house. You are usually told how long each person takes to paint a similarly-sized house, and you are asked how long it will take the two of them to paint the house when they work together.

Many of these problems are not terribly realistic — since when can two laser printers work together on printing one report? — but it's the technique that they want you to learn, not the applicability to "real life".

The method of solution for "work" problems is not obvious, so don't feel bad if you're totally lost at the moment. There is a "trick" to doing work problems: you have to think of the problem in terms of how much each person / machine / whatever does in a given unit of time . For instance:

Content Continues Below

## Suppose one painter can paint the entire house in twelve hours, and the second painter takes eight hours to paint a similarly-sized house. How long would it take the two painters together to paint the house?

To find out how much they can do together per hour , I make the necessary assumption that their labors are additive (in other words, that they never get in each other's way in any manner), and I add together what they can do individually per hour . So, per hour, their labors are:

But the exercise didn't ask me how much they can do per hour; it asked me how long they'll take to finish one whole job, working togets. So now I'll pick the variable " t " to stand for how long they take (that is, the time they take) to do the job together. Then they can do:

This gives me an expression for their combined hourly rate. I already had a numerical expression for their combined hourly rate. So, setting these two expressions equal, I get:

I can solve by flipping the equation; I get:

An hour has sixty minutes, so 0.8 of an hour has forty-eight minutes. Then:

They can complete the job together in 4 hours and 48 minutes.

The important thing to understand about the above example is that the key was in converting how long each person took to complete the task into a rate.

hours to complete job:

first painter: 12

second painter: 8

together: t

Since the unit for completion was "hours", I converted each time to an hourly rate; that is, I restated everything in terms of how much of the entire task could be completed per hour. To do this, I simply inverted each value for "hours to complete job":

completed per hour:

Then, assuming that their per-hour rates were additive, I added the portion that each could do per hour, summed them, and set this equal to the "together" rate:

As you can see in the above example, "work" problems commonly create rational equations . But the equations themselves are usually pretty simple to solve.

## One pipe can fill a pool 1.25 times as fast as a second pipe. When both pipes are opened, they fill the pool in five hours. How long would it take to fill the pool if only the slower pipe is used?

My first step is to list the times taken by each pipe to fill the pool, and how long the two pipes take together. In this case, I know the "together" time, but not the individual times. One of the pipes' times is expressed in terms of the other pipe's time, so I'll pick a variable to stand for one of these times.

Since the faster pipe's time to completion is defined in terms of the second pipe's time, I'll pick a variable for the slower pipe's time, and then use this to create an expression for the faster pipe's time:

slow pipe: s

together: 5

Next, I'll convert all of the completion times to per-hour rates:

Then I make the necessary assumption that the pipes' contributions are additive (which is reasonable, in this case), add the two pipes' contributions, and set this equal to the combined per-hour rate:

multiplying through by 20 s (being the lowest common denominator of all the fractional terms):

20 + 25 = 4 s

45/4 = 11.25 = s

They asked me for the time of the slower pipe, so I don't need to find the time for the faster pipe. My answer is:

The slower pipe takes 11.25 hours.

Note: I could have picked a variable for the faster pipe, and then defined the time for the slower pipe in terms of this variable. If you're not sure how you'd do this, then think about it in terms of nicer numbers: If someone goes twice as fast as you, then you take twice as long as he does; if he goes three times as fast as you, then you take three times as long as him. In this case, if he goes 1.25 times as fast, then you take 1.25 times as long. So the variables could have been " f  " for the number of hours the faster pipe takes, and then the number of hours for the slower pipe would have been " 1.25 f  ".

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## Math Work Problems - Two Persons

In these lessons, we will learn how to solve work problems that involve two persons who may work at different rates.

Related Pages Work Problems Solving Work Word Problems Using Algebra More Algebra Lessons

Work Problems are word problems that involve different people doing work together but at different rates . If the people were working at the same rate then we can use the Inversely Proportional Method instead.

## How To Solve Work Problems: Two Persons, Unknown Time

We will learn how to solve math work problems that involve two persons. We will also learn how to solve work problems with unknown time.

The following diagram shows the formula for Work Problems that involve two persons. Scroll down the page for more examples and solutions on solving algebra work problems.

This formula can be extended for more than two persons .

## "Work" Problems: Two Persons

Example: Peter can mow the lawn in 40 minutes and John can mow the lawn in 60 minutes. How long will it take for them to mow the lawn together?

Solution: Step 1: Assign variables : Let x = time to mow lawn together.

Step 3: Solve the equation The LCM of 40 and 60 is 120 Multiply both sides with 120

Answer: The time taken for both of them to mow the lawn together is 24 minutes.

## Work Problems With One Unknown Time

• Catherine can paint a house in 15 hours. Dan can paint a house in 30 hours. How long will it take them working together.
• Evan can clean a room in 3 hours. If his sister, Faith helps, it takes them two and two-fifths hours. How long will it take Faith working alone?

## Variations Of GMAT Combined Work Problems

• Working at a constant rate, Joe can paint a fence in 4 hours. Working at a constant rate, his brother can paint the same fence in 2 hours. How long will it take them to paint the fence if they both work together at their respective constant rates?
• Working alone at a constant rate, machine A takes 2 hours to build a care. Working alone at a constant rate, machine B takes 3 hours to build the same car. If they work together for 1 hour at their respective constant rates and then machine B breaks down, how much additional time will it take machine A to finish the car by itself?
• Working alone at a constant rate, Carla can wash a load of dishes in 42 minutes. If Carla works together with Dan and they both work at constant rates, it takes them 28 minutes to wash the same load of dishes. Working at a constant rate, how long would it take Dan to wash the load of dishes by himself?

## How To Solve “Working Together” Problems?

Example: It takes Andy 40 minutes to do a particular job alone. It takes Brenda 50 minutes to do the same job alone. How long would it take them if they worked together?

## Word Problem: Work, Rates, Time To Complete A Task

We are given that a person can complete a task alone in 32 hours and with another person they can finish the task in 19 hours. We want to know how long it would take the second person working alone.

Example: Latisha and Ricky work for a computer software company. Together they can write a particular computer program in 19 hours. Latisha van write the program by herself in 32 hours. How long will it take Ricky to write the program alone?

Latest news.

## Judicial Discretion in Cook County’s Problem-Solving Courts

Cook County’s problem-solving courts (PSCs) are specialized diversion courts that work with accused people whose mental health conditions and/or substance usage may have contributed to their arrests. The PSCs include Drug Treatment Courts, Veterans Treatment Courts, and Mental Health Courts; in other words, each type of PSC has a different target audience, all with the purported goal of rehabilitating individuals who meet the criteria for entry.

The Cook County PSCs have achieved some success in providing alternatives to incarceration. However, the research of the Collaboration for Justice of Chicago Appleseed Center for Fair Courts and the Chicago Council of Lawyers has recently raised concerns about some aspects of the PSCs that need to be addressed.

One issue of note are the low overall graduation rates of the PSCs and the especially low graduation rates of many individual courtrooms. The large variance in graduation rates across problem-solving courtrooms highlights the role of judicial discretion as an influence on success rates. Judges in PSCs hold significant discretion in determining the practices in their courtrooms, which ultimately affects participants’ outcomes.

Here, we discuss the scope and impact of judicial discretion in these specialty courts and why it deserves attention.

Cook County has 21 post-plea PSCs, which are categorized as Drug Treatment Courts (DTCs), Mental Health Courts (MHCs), or Veterans Treatment Courts (VTCs). The seven DTCs deal with accused people who use substances, the eight MHCs involve people with mental health concerns, and the six VTCs serve United States Veterans involved in the criminal legal system. Each PSC has different qualifications for entry, but generally, people enter these courts after they have pled guilty to charges that a court has identified as related to one of the aforementioned conditions. A participant “graduates” from a PSC if the judge certifies that they have met the requirements for that PSC.

The overall combined completion rate of all programs is 55% [but] there are even wider variations in graduation rates when comparing individual courtrooms. In general, each courtroom has a different judge. Because the profiles of PSC participants are unlikely to change much between programs with the same focus and procedures, the wide variation in graduation rates may suggest that judges have different practices for how people are terminated from PSC programs and for what reasons. The result is that it can be an accident of geography whether someone is in a program that has a higher graduation rate or a lower one. For example, someone who qualifies for MHC in Cook County’s Rolling Meadows branch court enters a program where 58% of participants succeed, whereas a person who qualifies at Chicago’s felony courthouse enters a program where only 41% succeed (“ One Size Doesn’t Fit All ” – March 2023).

The overall graduation rate of the post-plea PSCs is quite low – around 55% – but graduation rates vary across each PSC type: For the DTCs, the average graduation rate is 42%, participants in the MTCs graduate at a rate of 47%, and the VTCs have the highest graduation rate at 61%.

The relatively low graduation rates of the PSCs is of special concern because failure to graduate from a PSC significantly impacts participants’ lives going forward. For these post-plea programs, those who fail to graduate must fulfill the remainder of their sentence starting from the time they were admitted to a PSC; regardless of the amount of time they spent enrolled in the PSC, that time does not count toward their sentence. Thus, such participants sometimes spend more time engaged in the criminal legal system than they would have if they did not enter the PSCs at all. The cost for “failure” includes having a conviction on your record, which then limits housing, employment, and educational opportunities as a result of restrictive Illinois statutes . This is especially concerning since people who are engaged in the criminal legal system tend to be from low-income, primarily Black neighborhoods, which are already disproportionately harmed by historically racist drug laws .

## The Impact of a Judge

Important to note is our finding that even greater variation in graduation rates exists across the individual problem-solving courtrooms, which each have a different judge.

• The seven drug treatment courtrooms have the lowest skewing rates of graduation, ranging from just 29% in the Access to Community Treatment (ACT) Court to a high of 46% in the others.
• The county’s eight mental health courts range in graduation rates from 41% to 58%.
• The veterans treatment courts have the highest rates of success, ranging between 44% to 82% across different courtrooms.

This variance in rates of success suggests judicial discretion as a key factor contributing to many participants’ success or failure to graduate from the PSCs. Courtroom assignments are an accident of geography for most participants, and yet it could have dramatic consequences for their likelihood to graduate or face incarceration.

Judges in the PSCs have much greater discretion than they tend to have in traditional courts. First, judges have great control over participants’ treatment plans, the frequency of their drug testing, and even their personal lives. This generally enables the enforcement of abstinence-only policies, which counter many substance use best practices and include a very high level of surveillance of participants’ personal relationships and activities. Moreover, judges determine their own responses to participants’ violations of program guidelines and treatment plans, which in practice are often overly punitive: in some cases, judges have adjusted treatment plans or even relied on incarceration as sanctions.

## Reframing the Role of a Specialty Court Judge

In recent decades, advocates for education reform and educational sociologists have shifted their focus from dropout rates as a broad concern to the systemic factors that often push particular students out of educational systems. This “push out” framework is useful for considering the roles that those in power in courtrooms play in the failure of many participants to graduate from PSCs. The wide range of graduation rates across individual courtrooms suggests that the decisions individual judges make regarding treatment plans and sanctions may, in fact, “push” certain participants out of the PSCs.

In our report , we presented recommendations for how PSC judges should approach their role as guided by public health best practices:

• Incarceration should be entirely avoided for PSC participants. As incarceration presents intense mental and physical health risks, especially to people with preexisting mental health concerns, no participant should be incarcerated before or during their participation in a PSC. In particular, judges should end their practice of using incarceration as a sanction for active participants who break their rules in a PSC.
• Judges should grant participants autonomy to determine their own treatment plans based on consultation with healthcare providers rather than court actors. They should ensure that participants have access to a variety of treatment options, such as medication-assisted treatment, 12-step programs, and harm reduction services. Treatment should serve a participant’s individual health needs and should never be used as punishment.
• Judges should, to the best of their ability, respect medical confidentiality. PSC participants should never be required to disclose confidential medical or treatment information to the court in order to participate in a PSC. In the Restorative Justice Community Courts (RJCC), another diversion program in the Circuit Court of Cook County, advocates successfully pushed for legislation (signed in 2021) that ensures information provided during RJCC processes “is privileged and cannot be referred to, used, or admitted in any civil, criminal, juvenile, or administrative proceeding unless the privilege is waived…by the party or parties protected by the privilege.”

Though the problem-solving courts in Cook County divert some participants from carceral outcomes and connect them with public health resources, the manner in which judges employ their great discretion pushes many participants out of the program and into the criminal legal system. The PSCs have a dangerously low overall graduation rate, and graduation rates across courtrooms vary significantly. Judicial discretion is not an inherently bad thing; as described above, discretion in certain problem-solving courtrooms (i.e., VTCs with 82% graduation rates) has clearly led to success for many participants. That said, discretion should be used as a mechanism of support – not of control, surveillance, or punishment. To the extent possible, judges should use their discretion to support participants and connect them with community-based resources, which are essential for individuals’ wellbeing.

## Read the report by Chicago Appleseed Center for Fair Courts and the Chicago Council of Lawyers to learn more about the PSCs, our interview and observational data, and our recommendations for improvement.

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Home » Article » How to Use Salesforce to Manage Your Documentation

• How to Use Salesforce to Manage Your Documentation

No matter the size of your organization, or how many admins you have on your team (if you have a team!), one of the last things you probably enjoy doing is writing documentation. I hear you—it’s hard to start writing documentation, know what to capture, and figure out where to put it so you’ll actually go back to it as a reference.

We’ve talked about the importance and value of documenting your changes , but have you ever considered where ? I’ll be the first one to admit that it’s really easy to become overwhelmed and distracted by all the different tools out there to capture system changes. And there’s a lot of great ideas and conversations around documenting in a place where your business works, but what if that place doesn’t exist yet? What if there are too many places and none really meet your needs? What if you don’t have any budget to purchase that beloved tool you’ve read rave reviews about? Are you doomed to be bound to Word documents or endless Powerpoint slides? I say, no!

We’re in the business of solving problems for our businesses , so what if we built for ourselves the solution we need? What if you built your documentation repository right in Salesforce where you already work? I know, it’s an exciting idea, so let’s see an example in action.

## Why consider Salesforce for documentation?

This idea is not new or original, but it’s often overlooked because there are so many different tools out there. Here’s why you should consider using Salesforce for documentation.

Maybe you’re just starting out with a new org. Maybe you inherited an org and there’s zero documentation (what admin doesn’t know that story?). Maybe your department doesn’t have the budget for an additional tool. Or maybe the current tool doesn’t really fit your needs and you want something different.

Building a custom app to track change requests and enhancements is no different than building a custom app for a business process. And you already know about the cool automation you can build for yourself—and validation rules. 😉

## You can customize it for your needs

No paid solution is ever perfect, and even those that allow you to customize a solution start to feel like a side gig to your main gig. Building your own solutions means you can start with the fields/information you need to capture now and not worry about conforming to other processes that just don’t fit. And bonus: Using Salesforce to track and document the system changes you make allows you to stay in one place while you work. Who doesn’t love that?!

## It grows with you

To quote author Marshall Goldsmith, “What got you here won’t get you there.” The same logic applies here. Things you needed to track when you started will and should change as you grow. You won’t get it perfect on the first try, and that’s okay. Having your own custom app to track changes means you can change your mind and add/change fields and add apps (woah!) to further organize your work. Who doesn’t love scalability?

Pro-tip: Don’t try to think of everything on day one—you’ll risk over-engineering it for yourself. Aim for the 80/20 rule. If you can get 80% of what you need on day one, you can iterate and improve for that 20% that might look different as your team and business grows.

## You can easily share your work queue

Building your solution in Salesforce makes it a lot easier to share your workload with requestors, managers, and others. Expose as much or as little as you want with our friend Field-Level Security, and inform people where their request is in the queue without having to reply to an email, Slack, Teams notice, or the carrier pigeon they asked you with. And when something is done, you’ve got a great platform to announce what’s gone live.

## You get awesome analytics

Raise your hand if you were waiting for this one! 🙂 Yup, there’s power in numbers. Building your own custom fields and objects allows you to also capture the details you need to answer questions like:

• How many new features did I/we deploy this month/quarter/year?
• How many could I/we not get to? (smells like a great metric to advocate for headcount)
• How many bug fixes did we do?
• What’s the average time it takes to complete this type of request? (Ooooo, don’t you love that for future planning?!)
• Which department/person/team is consuming the most of my/our time?
• Are we fairly dividing our time across teams/departments?
• Which features are must-haves versus nice-to-haves?

You are only limited by your imagination! So, instead of “Why should I use Salesforce to track my changes?”, I hope you’re thinking, “Why shouldn’t I use Salesforce?!”

## Example solve

Now, you might be wondering, “Those are pretty good reasons to consider using Salesforce for my documentation, but how does it come together?” I’m so glad you asked!

Like any business challenge presented to you, there’s always more than one way to solve it. For example, some folks love to use the Case object because so much comes out-of-the-box. Our custom app below, which we’ll call SFDC Requests, is just one example of the many great ways to use Salesforce for your documentation. As you read on, consider which elements work for you and your business .

## Decide what you need to track

Like any solve, you first have to determine not only what’s important to track for communication up and out about your work but also what’s helpful for you , #AwesomeAdmin, when you need to return to your past solve (or onboard a new admin!). Which feature/tool will you use to solve it (Flow, formulas, validation rules)? Who asked for it, and when is it needed by? How much time will it take, what’s the business case, and how will you solve it? Answering these questions helps you populate the list of fields to build.

Here’s an example of the intake details you can capture when creating a change request.

Built for efficiency, this quickly captures the key pieces you need to log a request so that you can both prioritize it and know the goal when you return to work on it.

1. Lightning is your friend! Use tabs to help break apart the process; for example, what’s needed upon creation versus what’s a resolution field.

a. Pro-tip: Create a tab that’s a text box where you can drop links or process related instructions to yourself or your team when they fill in information. (That’s the REQ Instructions tab in the screenshot above.)

2. Capture the subject of the ask for quick filtering in list views. Keep it obvious and full of keywords so you can leverage Salesforce Einstein Search.

3. Who asked for it? Great to know when you have follow-up questions.

4. Is this a feature you can improve or lean out in the future? Easily flag enhancements you know are quick fixes or that future features will improve.

5. Is this a new feature or a bug fix? (Can you smell the beautiful analytics now?)

6. When does the requestor need or wish to have it by? (Hello, prioritization!)

7. Maybe you’re not modifying Salesforce but another application like Account Engagement (aka Pardot—I’m with you, Pardashians). Who said this could only be used for Salesforce? 😉

8. What’s the business need and problem they’re trying to solve? Context is magic when you, and future admins, come back to this solve.

Here’s an example of the resolution details you can capture when completing a change request.

9. How urgent is this request above all others? This field can live anywhere and is best placed based on your business process for prioritizing requests that come in.

10. How long does it take (in the time interval you like) to complete the request? Be sure to define what goes in here. Do you include the meetings you sat through to discuss and test? Create test material? Train? Build? There’s no wrong answer except an inconsistent one, so define it for you and your team, and stick to it. This number is helpful when you need to advocate for more time on a solution or adding headcount!

11. Which objects are used? Despite how much flack multi-select picklists get, you , # AwesomeAdmin, know how to handle them. Here, you can list all of the active objects in your system, and you can flag which ones are modified in the solve. Read that again… Yup, imagine knowing which objects were impacted when you need to troubleshoot an issue.*

12. Do you have a team lead or a person responsible for the ask, but someone different executing on it? Having two fields can be helpful, but usually ‘owner’ is sufficient for who is managing the request.

13. When will/did you finish it? (And how did that compare to when they needed it? I know, I know! The analytical possibilities are endless!)

14. Another multi-select picklist of the different tools you can use to solve the request. Is it Flow? A validation rule? Why type it when you can select it?*

*See the additional screenshot of numbers 11 & 14 for examples of the information you can pre-fill for easy tracking.

Hopefully, the above fields got you thinking about all the great ways you can easily and quickly capture for your change requests. But there’s always more. Here are some pro-tips to take your documentation to the next level!

## Pro-tips (because you’re awesome, and your solves are too)

• Use paths to easily identify where the request is in the workflow.
• Add additional features like Tasks, Topics, Notes, and more to set reminders for yourself and capture meeting minutes. (Woah, what a great use case for Notes!)
• Got Slack? For those who use Slack and sometimes need to vet a request’s impact before building it, a fun way to share the love is to build in a ‘post to Slack’ screen flow. This screen flow is built to push the request link to a team channel for input before building. Once the message is sent, the screen flow updates with a hyperlink to the very same thread in Slack to both know you asked and also jump right to that thread for feedback details.
• Running projects and not just managing system changes? Consider downloading the free PMT Project app from Salesforce Labs. You can customize this app and even connect your change requests’ custom object to larger initiatives. For example, onboarding a new department or process? Create a project with the app to document the large effort of migrating a department or process, and keep track of the system changes you need to make for the overall project by linking those change requests to the project. It’s a great way to organize larger efforts that have many moving parts. Learn more about PMT Projects here .

Hopefully, now you’re all in on the awesome ideas and ways to build your own documentation app in Salesforce. You’ve seen a lot of fields that help you better organize your work and also have some great details when you return to a solve. So, what does this information offer you when it comes to answering some of those big questions we asked earlier? Here’s just an example, but any field—yes, even your time fields—can be reported on! Again, you’re only limited by your imagination and business needs.

## Get building!

Now it’s your turn to build the documentation app of your dreams, right inside of Salesforce. Remember, this app is for you/your team to not just better organize and capture what you’ve done, but to do so in a fun way that meets your business style and need. We’re huge advocates of Salesforce for the rest of our business—let’s give ourselves a win too. Have fun and maybe even treat yourself to some screen flows, action buttons, and ‘Ooooh, I always forget that field’ validation rules.

So, what are you waiting for?! Get building! 😀

• External Site: Salesforce Ben: Complete Guide to Salesforce Documentation (in an AI World)
• AppExchange: Project Management Tool – PMT

## Jennifer Cole

Jennifer Cole is a Salesforce Admin with 10+ years of experience as both an administrator and business analyst. She is 4x certified and a Ranger with over 100 badges.

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## How to Write Great Documentation to Help with Future Problem Solving

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## Introducing Dynamic Interactions, the Latest Low-Code Innovation for Salesforce Platform

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## Have an Idea for a Story?

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## How Covid symptoms have evolved since the pandemic

Doctors say they're finding it increasingly difficult to distinguish Covid from allergies or the common cold, even as hospitalizations tick up .

The illness' past hallmarks, such as a dry cough or the loss of sense of taste or smell, have become less common. Instead, doctors are observing milder disease, mostly concentrated in the upper respiratory tract.

"It isn’t the same typical symptoms that we were seeing before. It’s a lot of congestion, sometimes sneezing, usually a mild sore throat," said Dr. Erick Eiting, vice chair of operations for emergency medicine at Mount Sinai Downtown in New York City.

The sore throat usually arrives first, he said, then congestion.

The Zoe COVID Symptom Study, which collects data on self-reported symptoms in the U.K. through smartphone apps, has documented the same trend . Its findings suggest that a sore throat became more common after the omicron variant grew dominant in late 2021. Loss of smell, by contrast, became less widespread, and the rate of hospital admissions declined compared to summer and fall 2021.

Doctors now describe a clearer, more consistent pattern of symptoms.

"Just about everyone who I've seen has had really mild symptoms," Eiting said of his urgent care patients, adding, "The only way that we knew that it was Covid was because we happened to be testing them."

## How do Covid symptoms progress?

Though three doctors interviewed said Covid commonly begins with a sore throat these days, they gave differing descriptions of the severity.

Dr. Grace McComsey, vice dean for clinical and translational research at Case Western University, said some patients have described "a burning sensation like they never had, even with strep in the past."

"Then, as soon as the congestion happens, it seems like the throat gets better," she said.

Along with congestion, doctors said, some patients experience a headache, fatigue, muscle aches, fever, chills or post-nasal drip that may lead to a cough — though coughing isn't a primary symptom.

McComsey said fatigue and muscle aches usually last a couple of days, whereas congestion can sometimes last a few weeks.

She estimated that only around 10-20% of her Covid patients lose their sense of taste or smell now, compared to around 60-70% early in the pandemic.

Eiting said he's not seeing a lot of diarrhea lately, either — a more common symptom in the past.

For the most part, the doctors said, few patients require hospitalization — even those who show up at emergency rooms — and many recover without needing the antiviral pill Paxlovid or other treatment.

"Especially since July, when this recent mini-surge started, younger people that have upper respiratory symptoms — cough, runny nose, sore throat, fever and chills — 99% of the time they go home with supportive care," said Dr. Michael Daignault, an emergency physician at Providence Saint Joseph Medical Center in Burbank, California.

## Why Covid seems milder now

Dr. Dan Barouch, director of the Center for Virology and Vaccine Research at Beth Israel Deaconess Medical Center in Boston, attributed the mild symptoms that doctors are seeing to immunity from vaccines and previous infections.

"Overall, the severity of Covid is much lower than it was a year ago and two years ago. That’s not because the variants are less robust. It’s because the immune responses are higher," Barouch said.

Other doctors think that omicron itself also changed the presentation of Covid symptoms, since some studies have shown that early versions of it weren’t as good as previous variants at infecting the lungs .

The most prevalent subvariant circulating now is EG.5, followed by a strain called FL.1.5.1. Together, those two appear to be driving an uptick in Covid infections, though scientists are also watching BA.2.86 , a variant with a large number of mutations that looks significantly different from previous versions of omicron. Though cases of BA.2.86 are rising in the U.S., it isn’t among the top variants circulating.

Barouch said the new booster shots should be effective against those three strains and others.

## Who is being hospitalized?

The U.S. is recording around 19,000 Covid hospitalizations per week, according to the Centers for Disease Control and Prevention. The weekly average rose around 80% from early August to the beginning of September.

Hospitalization rates are highest among people ages 75 and up, followed by babies under 6 months and adults ages 65 to 74. Most people hospitalized for Covid since January had not received a bivalent booster, according to the CDC.

Older people in particular may have waning immunity if they haven’t been infected or vaccinated recently, Daignault said.

"That’s why the priority should be to vaccinate that particular group of patients with the fall booster," he said.

Daignault said emergency rooms generally aren’t seeing the shortness of breath, low oxygen rates or viral pneumonia that led some patients to be put on oxygen tubes or ventilators in the past.

Instead, he said, the typical Covid patients hospitalized in Burbank are older and suffering dehydration, loss of appetite, weakness or fatigue.

## What does long Covid look like now?

A study published this month found that long Covid rates declined once omicron became the dominant variant. Researchers don’t know if milder disease contributed to that trend, or if population immunity was largely responsible.

But McComsey — a principal investigator for the National Institutes of Health’s RECOVER Initiative, which studies long Covid — said she's still seeing new cases of long Covid. Rapid heart rate and exercise intolerance are among the most common lingering symptoms, she said.

Each re-infection brings a risk of long Covid, McComsey added, so she doesn't think people should ignore the current rise in infections.

"What we’re seeing in long Covid clinics is not just the older strains that continue to be symptomatic and not getting better — we’re adding to that number with the new strain as well," McComsey said. "That’s why I’m not taking this new wave any less seriously."

Aria Bendix is the breaking health reporter for NBC News Digital.

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• Fact Sheets

## Fact Sheet: The Biden-Harris Administration Takes New Actions to Increase Border Enforcement and Accelerate Processing for Work Authorizations, While Continuing to Call on Congress to Act

WASHINGTON – Today, the Department of Homeland Security (DHS) is announcing another series of actions to increase enforcement across the Southwest Border, accelerate processing of work authorizations, and the decision to redesignate and extend Temporary Protected Status (TPS) for Venezuela. In consultation with interagency partners and with careful consideration for the conditions, and due to extraordinary and temporary conditions in Venezuela that prevent individuals from safely returning, the Secretary of Homeland Security decided to extend and redesignate TPS for Venezuela. President Biden has called on Congress to pass comprehensive immigration reform since his first day in office. As a result of Congress’ failure to enact the reform, the Administration has been using the limited tools it has available to secure the border and build a safe, orderly, and humane immigration system while leading the largest expansion of lawful pathways for immigration in decades. We also urge passage of fully funded emergency appropriations, including the supplemental funding request for border security, as requested by the President this summer. The $4 billion supplemental funding requested for DHS addresses immediate needs of the Department to safely and humanely manage the Southwest Border and to continue implementing our immigration laws through the expansion of lawful pathways and enforcing consequences for those who do not use them. DHS has also allocated more than$770 million to 69 partner organizations in Fiscal Year 2023 to support communities receiving migrants, in both the Southwest Border region and the interior, through the Shelter and Services Program (SSP) and the Emergency Food and Shelter Program – Humanitarian Awards (EFSP-H). Combating smugglers. DHS continues to escalate the fight against those smuggling in persons and narcotics and the Administration is prosecuting an increasing number of smugglers, as well as noncitizens who are violating our laws.

From April 2022 through September 12, 2023, CBP and HSI arrested nearly 17,000 suspected human smugglers and seized more than $51 million in property and nearly$13 million in currency. This has resulted in more than 2,000 indictments and more than 1,500 convictions in partnership with the U.S. attorneys.

U.S. Border Patrol has referred 9,904 individuals for prosecution between May 12 and Sept. 14.

The Department of Justice (DOJ) is vigorously prosecuting those who unlawfully bring in, harbor, or transport migrants, as well as many thousands of felony reentry cases. DOJ and DHS are working closely together to target additional prosecutorial resources towards these serious immigration offenses.

• Border Security
• Citizenship and Immigration Services
• Immigration and Customs Enforcement
• Customs and Border Protection (CBP)
• Immigration
• Immigration and Customs Enforcement (ICE)
• Lawful Pathways
• Southwest Border
• U.S. Citizenship and Immigration Services (USCIS)

#### IMAGES

1. Solving Work Rate Problems

2. Work Rate Problems

3. NTI Day #9

4. Solving Applied Problems Involving Rate, Time and Work

5. Unit Rate Problem Solving Graphic Organizer for 6th Grade

6. Work & Rate: Understanding & Solving Combined Rate Problems in Less than 1 Minute (No Formulas)

#### VIDEO

1. Neon

2. 4. Related Rates

3. Initial Rate Method

4. Solving a distance, rate, time problem using a rational equation

5. Solving Work and Rate Word Problems with Ease

6. Math Constant Rate and Percentage Problem

1. Work Rate Problems with Solutions

Work Rate Problems with Solutions Problem 1: It takes 1.5 hours for Tim to mow the lawn. Linda can mow the same lawn in 2 hours. How long will it take John and Linda, work together, to mow the lawn? Solution to Problem 1: We first calculate the rate of work of John and Linda John: 1 / 1.5 and Linda 1 / 2

2. 9.3: Work-rate problems

Notice the time given doing the job together: 1 1 hour and 12 12 minutes. Unfortunately, we cannot use this format in the work-rate equation. Hence, we need to convert this to the same time units: 1 1 hour and 12 12 minutes = 112 60 = 1 12 60 hours = 1.2 = 1.2 hours = 6 5 = 6 5 hours. Table 9.3.5. time.

3. Lesson HOW TO Solve Rate of Work (painting, pool filling, etc) Problems

Basic Formula Let's assume we have two workers (pipes, machines, etc): A and B. Worker A can finish a job in X hours when working alone. Worker B can finish a job in Y hours when working alone The number of hours they need to complete the job when they're both working at the same time is given by

4. PDF SOLVING WORK-RATE PROBLEMS

SOLVING WORK-RATE PROBLEMS Part I: Introduction To solve work-rate problems it is helpful to use a variant of distance equals rate times time. Specifically: Q rt In this formula Q is the quantity or amount of work done, r is the rate of work and t is the time worked. EX 1: If a machine can produce 2 1 parts per minute then in: 2

5. 9.10 Rate Word Problems: Work and Time

The equation to solve is: 1 R + 1 2R = 1 10 h, where Doug's rate ( 1 D) = 1 2 × Becky's ( 1 R) rate. Sum the rates: 1 R + 1 2R = 2 2R + 1 2R = 3 2R Solve for R: 3 2R = 1 10 h which means 1 R = 1 10 × 2 3 h so 1 R = 2 30 or R = 30 2 1 R + 1 2 R = 1 10 h, where Doug's rate ( 1 D) = 1 2 × Becky's ( 1 R) rate.

6. 3 Ways to Solve Combined Labor Problems

The problems basically require finding unit rates, combining them, and setting them equal to an unknown rate. These problems require a lot of interpretive logic, but as long as you know how to work with fractions, solving them is fairly easy.

7. Word Problems: Rate of work, PAINTING, Pool Filling Lessons

Algebra Rate-of-work-word-problems. Word Problems: Rate of work, PAINTING, Pool Filling. Solvers. Lessons. Answers archive. Lesson : Using fractions to solve word problems on joint work by ikleyn (48869) Lesson : Solving rate of work problem by reducing to a system of linear equations by ikleyn (48869) Lesson : Using quadratic equations to ...

8. Lesson 23: Problem Solving Using Rates, Unit Rates, and Conversions

Lesson 23: Problem Solving Using Rates, Unit Rates, and Conversions . Student Outcomes Students solve constant rate work problems by calculating and comparing unit rates. Materials Calculators . Classwork (30 minutes) If work is being done at a constant rate by one person at a different constant rate by another person, both

9. Lesson Solving rate of work problem by reducing to a system of linear

Let m be the the man's rate of work and w be the woman's rate of work. Then we have the linear system of two equations in two unknowns, in accordance with the given data: , or. . To solve it, subtract the second equation of the last system from the first equation. You will get. 6m = 10w, or w = = . Next, substitute it into the first equation of ...

10. Rate problems (video)

Lesson 1: Intro to rates Rate review Math > 6th grade > Rates and percentages > Intro to rates © 2023 Khan Academy Cookie Notice Rate problems Google Classroom About Transcript In this math lesson, we learn to find unit rates and use them to solve problems. We first calculate the rate for one unit, like cars washed per day or cost per battery.

11. Work Problems

Learn How to Solve Rational Equations Hone Problem Solving Skills and Mathematical Reasoning Solve Real World Time-Management Problems Work Basics There is a basic principle related to work problems that must be dealt with. It is the formulation of work in a given unit of time. This question will demonstrate the need to know this basic skill:

12. Rate problems (practice)

Lesson 1: Intro to rates Rate problems Google Classroom You might need: Calculator Lynnette can wash 95 95 cars in 5 5 days. How many cars can Lynnette wash in 11 11 days? cars Show Calculator Stuck? Review related articles/videos or use a hint. Report a problem 7 4 1 x x y y \theta θ \pi π 8 5 2 0 9 6 3 Do 4 problems

13. Rational equations word problem: combined rates

I just can't figure out why one rate (hour/lawn) is incorrect to solve the problem while going with another rate (lawn/hour) is correct. ... the rates into (y/x part of the work done / (per) time interval notation) which in your particular problem would be 13/40 (combined efforts) / (per) day. It may seem not that intuitive but 13/40 represents ...

14. Rate problems 2 (practice)

Course: Algebra (all content) > Unit 15. Lesson 3: Word problems with multiple units. Worked example: Rate problem. Rate problems 2. Multiple units word problem: road trip. Measurement word problem: running laps. Using units to solve problems. Using units to solve problems: Drug dosage. Math >.

15. GMAT Work Rate Problems

The questions will often give you information about times and about amounts, and what you need to know is: you can't add or subtract times to complete a job and you can't add or subtract amounts of work; instead, you add and subtract rates. (rate of A alone) + (rate of B alone) = (combined rate of A & B)

16. "Work" Word Problems

The method of solution for "work" problems is not obvious, so don't feel bad if you're totally lost at the moment. There is a "trick" to doing work problems: you have to think of the problem in terms of how much each person / machine / whatever does in a given unit of time. For instance: MathHelp.com

17. Rate and Work Problems Flashcards

Key to solving Rate/Work Problems - Identify the formula that corresponds to the question - Identify what formula component you are asked to solve for - Keep track of units and convert to the appropriate unit before answering the question. Combined Work Formula - give you 2 or more individual work rates then ask you for the combined rate ...

18. Work Word Problems (video lessons, examples, solutions)

Step 1: Assign variables: Let x = time to mow lawn together. Step 2: Use the formula: Step 3: Solve the equation The LCM of 40 and 60 is 120 Multiply both sides with 120 Answer: The time taken for both of them to mow the lawn together is 24 minutes. Example 2: It takes Maria 10 hours to pick forty bushels of apples.

19. Rate Work Problems

Students solve constant rate work problems by calculating and comparing unit rates. Lesson 23 Summary. Constant rate problems always count or measure something happening per unit of time. The time is always in the denominator. Sometimes the units of time in the denominators of two rates are not the same. One must be converted to the other ...

20. Math Work Problems (video lessons, examples and solutions)

Solution: Step 1: Assign variables: Let x = time to mow lawn together. Step 2: Use the formula: Step 3: Solve the equation The LCM of 40 and 60 is 120 Multiply both sides with 120 Answer: The time taken for both of them to mow the lawn together is 24 minutes. Work Problems With One Unknown Time Examples: Catherine can paint a house in 15 hours.

21. GRE Quantitative: Rates and Work Question Practice

The problem asks how many miles the trail was one way. 8.4 / 2 = 4.2. The answer to the question is 4.2 miles. You could also solve this problem in other ways, including using a system of equations and substitution, but it's nice to know that you can pick a number for the Distance traveled and use it to find the Average Rate for the whole ...

22. Solving Problems Using Rates

Under the time column, we will put 18 for Brother 1 since the problem states he can only work for 18 hours. ... To solve rate word problems, you first set up a table following your rate formula ...

23. Solving Unit Rate Word Problems

Welcome to Solving Unit Rate Word Problems with Mr. J! Need help with unit rates? You're in the right place!Whether you're just starting out, or need a quick...

24. Judicial Discretion in Cook County's Problem-Solving Courts

The PSCs have a dangerously low overall graduation rate, and graduation rates across courtrooms vary significantly. Judicial discretion is not an inherently bad thing; as described above, discretion in certain problem-solving courtrooms (i.e., VTCs with 82% graduation rates) has clearly led to success for many participants.

25. How to Use Salesforce to Manage Your Documentation

a. Pro-tip: Create a tab that's a text box where you can drop links or process related instructions to yourself or your team when they fill in information. (That's the REQ Instructions tab in the screenshot above.) 2. Capture the subject of the ask for quick filtering in list views.

26. Covid symptoms are now more mild and follow a pattern, doctors say

Loss of smell, by contrast, became less widespread, and the rate of hospital admissions declined compared to summer and fall 2021. Doctors now describe a clearer, more consistent pattern of symptoms.

27. Fact Sheet: The Biden-Harris Administration Takes New Actions to

The Department of Homeland Security (DHS) is announcing another series of actions to increase enforcement across the Southwest Border, accelerate processing of work authorizations, and the decision to redesignate and extend Temporary Protected Status (TPS) for Venezuela.